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Properties

Label 2-29-29.28-c13-0-20
Degree $2$
Conductor $29$
Sign $0.942 + 0.334i$
Analytic cond. $31.0969$
Root an. cond. $5.57646$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 67.0i·2-s + 954. i·3-s + 3.70e3·4-s + 4.83e4·5-s + 6.39e4·6-s + 4.44e5·7-s − 7.97e5i·8-s + 6.83e5·9-s − 3.24e6i·10-s + 3.48e6i·11-s + 3.53e6i·12-s − 7.24e6·13-s − 2.98e7i·14-s + 4.61e7i·15-s − 2.31e7·16-s − 8.18e7i·17-s + ⋯
L(s)  = 1  − 0.740i·2-s + 0.755i·3-s + 0.451·4-s + 1.38·5-s + 0.559·6-s + 1.42·7-s − 1.07i·8-s + 0.428·9-s − 1.02i·10-s + 0.593i·11-s + 0.341i·12-s − 0.416·13-s − 1.05i·14-s + 1.04i·15-s − 0.344·16-s − 0.822i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.942 + 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.942 + 0.334i$
Analytic conductor: \(31.0969\)
Root analytic conductor: \(5.57646\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :13/2),\ 0.942 + 0.334i)\)

Particular Values

\(L(7)\) \(\approx\) \(3.708974053\)
\(L(\frac12)\) \(\approx\) \(3.708974053\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (3.01e9 + 1.07e9i)T \)
good2 \( 1 + 67.0iT - 8.19e3T^{2} \)
3 \( 1 - 954. iT - 1.59e6T^{2} \)
5 \( 1 - 4.83e4T + 1.22e9T^{2} \)
7 \( 1 - 4.44e5T + 9.68e10T^{2} \)
11 \( 1 - 3.48e6iT - 3.45e13T^{2} \)
13 \( 1 + 7.24e6T + 3.02e14T^{2} \)
17 \( 1 + 8.18e7iT - 9.90e15T^{2} \)
19 \( 1 - 2.18e8iT - 4.20e16T^{2} \)
23 \( 1 + 4.04e8T + 5.04e17T^{2} \)
31 \( 1 + 3.15e9iT - 2.44e19T^{2} \)
37 \( 1 + 1.36e10iT - 2.43e20T^{2} \)
41 \( 1 + 2.51e10iT - 9.25e20T^{2} \)
43 \( 1 - 2.89e10iT - 1.71e21T^{2} \)
47 \( 1 - 9.13e10iT - 5.46e21T^{2} \)
53 \( 1 - 2.11e11T + 2.60e22T^{2} \)
59 \( 1 + 1.71e11T + 1.04e23T^{2} \)
61 \( 1 + 4.80e10iT - 1.61e23T^{2} \)
67 \( 1 - 1.56e11T + 5.48e23T^{2} \)
71 \( 1 + 5.26e11T + 1.16e24T^{2} \)
73 \( 1 - 1.93e12iT - 1.67e24T^{2} \)
79 \( 1 + 7.46e11iT - 4.66e24T^{2} \)
83 \( 1 + 3.72e12T + 8.87e24T^{2} \)
89 \( 1 - 7.80e12iT - 2.19e25T^{2} \)
97 \( 1 + 3.28e12iT - 6.73e25T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.10128619865148698122860060520, −12.61634201688410585983796769927, −11.30286281294520547334022153926, −10.20201779628884345966285881729, −9.510210750808291668504821513729, −7.42210670922264374958149793457, −5.60649845529241773832751531978, −4.23383556359686852021779327152, −2.25969278580939687461221289839, −1.47145101175029839334795912665, 1.44401234491134092381132060773, 2.20184341675218975257649618974, 5.09691929746897717261718906765, 6.22421927352558550196671723424, 7.38255493531577503934113624558, 8.582342507451905606344696936935, 10.43107736707469107936179334945, 11.70463433696455469468842670982, 13.27484455991508031537535359447, 14.20870684110166874307794216619

Graph of the $Z$-function along the critical line